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It is given that the probability that one among two radioactive atoms will decay in an interval $x,x+dx$ and another one will decay in a time interval $y+dy$ is given by the following expression: $$Ne^{-y\alpha-x\beta-\gamma\sqrt{xy}}dydx$$ where $\alpha, \beta>0$, and $\gamma~\epsilon~R$.

What would be the normalization factor, $N$ of the probability density function in the analytical form when it is defined only in the first quadrant? Also, how do I numerically calculate the approx. probability that one of them decays prior to the other?

(Please do show me how to solve the integral)

Spoilt Milk
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1 Answers1

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N must be such that the integral, taking x and y each from 0 to infinity is 1. In other words, integrate $e^{-y\alpha- x\beta- \gamma\sqrt{xy}}dydx$ from with x and y from 0 to infinity, $\int_0^\infty\int_0^\infty e^{-y\alpha- x\beta- \gamma\sqrt{xy}}dydx$, and N is 1 over that. The probability that the second atom decays before the first is the integral with x going from 0 to infinity and y from 0 to x: $N\int_0^\infty\int_0^x e^{-y\alpha- x\beta- \gamma\sqrt{xy}}dydx$

user247327
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  • Thanks for the answer! Although I knew the method of solving, I was unsuccessful in obtaining a definite answer. Also, how can I determine the expected value and standard deviation of decay time for both atoms in analytical form? – Spoilt Milk Oct 27 '17 at 12:39